Muon g-2 Predictions Recapped

A new 192 page paper prepared by a huge collaboration of authors and led by Fermilab exhaustively reviews the latest efforts to theoretically calculate the property of anomalous magnetic moment of the muon, called muon g-2 reports on the latest state of that effort. Another big review came out earlier this year in March.

The tension between experiment and theoretical prediction of this quantity is one of the leading unexplained problems in physics today. A September 9, 2019 post at this blog also examined the discrepancy in detail. As I summarized then, using slightly different source numbers from this paper (which, if anything assigns a larger share of the error to the QCD component):

The errors in theoretical calculation by component are summarized roughly as follows (in comparable units):

QED 0.08 (i.e. 0.2% of the total)
Weak Force 1.00 (i.e. 2.9% of the total)
QCD 33.73 (i.e. 96.9% of the total). . . .

Proportion of total value from each component:

QED 99.994% (116 584 718.95)
Weak Force 0.00013% (153.6)
QCD 0.006% (6931)

Relative error percentage:

QED 0.000 000 0686%
Weak Force 0.65%
QCD 4.88%

On one hand, muon g-2 is predicted to exquisite seven significant digit accuracy, suggesting the general correctness of the Standard Model used to calculate it. On the other hand, a strong tension with the most recent precise experimental measurement of that quantity, remains, dashing the hope that much of the discrepancy might have been due to uncertainties in the most uncertain part of the calculation. This review is timely because a new, four times more precise experimental measurement will be available in a year or two.  

Improvements of the precision in the theoretical estimate, which are predominantly from the quantum chromodynamics component of the calculation that make the smallest contribution to the absolute value of this measurable quantity are also expected in the near term.

Conventional wisdom is that the tension will ease, producing an experimentally measured value closer to the theoretically predicted one, despite the much greater precision of the new measurement. 

But, if this does not happen, the quantity muon g-2 is a good global indicator of the existence and magnitude of beyond the Standard Model physics, although it is not a very good tool for determining the precise nature of the new physics because it is affected by essentially all parts of the Standard Model when measured with the precision that is now possible.

A third possibility is that scientists are making some shared conceptual error in how they do their theoretical predictions of the value of muon g-2 that doesn't really amount to new physics. Gravitational corrections, if any, however, should be negligible in magnitude compared to other uncertainties. 

Cheat Sheet For When Experimental Results Are Announced

The current state of the art Brookhaven measurement was:

116,592,089(63) x 10^−11.

The headline that will be used when the experimentally measured value announced later this year or next year can be determined by referring the the cheat sheet below where the number listed is the new experimental result.

Less than or equal to 116,591,582 x 10^-11:

New physics has been discovered contradicting the Standard Model more definitively than any prior measurement in physics. The Brookhaven measurement was deeply flawed.

116,591,581 to 116,591,718 x 10^-11:

There is tension with the Standard Model prediction in the opposite direction from the Brookhaven measurement. The Brookhaven measurement was deeply flawed. At the high end of this range it will be called a "slight tension", at the low end of this range it will be called a "strong tension."

116,591,718 to 116,591,901 x 10^11:

The new measurement has confirmed the Standard Model prediction. Any new physics that impact muon g-2 are too small to observe experimentally.

116,591,902 to 116,592,037 x 10^11:

There is tension with the Standard Model prediction. At the low end of this range it will be called a "slight tension", at the high end of this range it will be called a "strong tension."

Greater than or equal to 116,592,038 x 10^11:

New physics has been discovered contradicting the Standard Model more definitively than any prior measurement in physics.

The Facts

The new paper and its abstract are as follows:

We review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. 

This is performed in a perturbative expansion in the fine-structure constant 

α and is broken down into pure QED, electroweak, and hadronic contributions. 

The pure QED contribution is by far the largest and has been evaluated up to and including (α5) with negligible numerical uncertainty. The electroweak contribution is suppressed by (mμ/MW)2 and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at (α2) and is due to hadronic vacuum polarization, whereas at (α3) the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads aSMμ=116591810(43)×10−11 and is smaller than the Brookhaven measurement by 3.7σ. 

The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future-which are also discussed here-make this quantity one of the most promising places to look for evidence of new physics.

T. Aoyama, et al., "The anomalous magnetic moment of the muon in the Standard Model" arXiv (June 8, 2020).

The conclusion is as follows:

In this paper we provide a detailed analysis and review of the SM calculation of the muon anomalous magnetic moment aµ. The emphasis is on the hadronic contributions, since they dominate the final uncertainty, but the QED and electroweak contributions are also discussed in detail and up-to-date numbers are provided. 

The QED contribution, which has been calculated up to tenth order in the perturbative expansion, i.e., O(α 5 ), is reviewed in Sec. 6. The final number depends on the input used for the fine-structure constant α and at present there are two independent determinations that differ by about 2.4 standard deviations. The impact of this discrepancy on the final number for aµ is however well below the uncertainty of the QED contribution itself, which is dominated by the estimated effect of the O(α 6 ) contribution. As final number we take the one based on the value of α obtained from atom-interferometry measurements of the Cs atom [117], see Eq. (6.30), and the latest QED calculations from Refs. [33, 34]: 

a QED µ (α(Cs)) = 116 584 718.931(104) × 10−11 . (8.1) 

Electroweak contributions are reviewed in Sec. 7: they have been calculated up to two loops and an estimate of the leading logarithmic contribution beyond two-loop level is also included in the final estimate. The hadronic loops, which appear at two-loop level, are also included and dominate the uncertainty of the EW contribution. The final result Eq. (7.16) (mainly based on Refs. [35, 36]) reads 

a EW µ = 153.6(1.0) × 10−11 , (8.2) 

with an uncertainty ten times larger than the QED one, but still negligible with respect to the hadronic uncertainties. 

In the section on data-driven evaluations of HVP we reviewed both the available data sets for the e + e − → hadrons cross section and the techniques applied for the evaluation of the HVP dispersive integral. In particular, we provide a detailed discussion of the differences between these approaches and the current limitations of the dispersive HVP evaluation, as they arise from the published experimental uncertainties as well as, crucially, from unresolved tensions among the data sets, especially in the dominant ππ channel. As the main result, Eq. (2.33), we devised a merging procedure that adequately takes into account these tensions, which also drive the differences between the available HVP evaluations. The resulting estimate, based on Refs. [2–7] as well as the main experimental input from Refs. [37– 89], 

a HVP, LO µ = 6931(40) × 10−11 (8.3) 

should provide a conservative but realistic assessment of the current precision of data-driven HVP evaluations. In the same framework, the LO result is complemented by NLO [7] and NNLO [8] HVP iterations, see Eq. (2.34) and Eq. (2.35), 

a HVP, NLO µ = −98.3(7) × 10−11 , 

a HVP, NNLO µ = 12.4(1) × 10−11 , (8.4) 

leading to the sum 

a HVP, LO µ + a HVP, NLO µ + a HVP, NNLO µ = 6845(40) × 10−11 . (8.5) 

Finally, we discussed the prospects for future improvements, including new data from several e + e − experiments as well as the possibility to measure HVP independently in electron–muon scattering. 

The status of lattice QCD+QED calculations of HVP is reviewed in Sec. 3. While lattice calculations can, in principle, provide an alternate, ab initio determination of the HVP contribution, they are, at present, not precise enough to confront the data-driven evaluations. The current “lattice world average,” obtained in Sec. 3.5.1 from a conservative combination of current, published lattice QCD+QED results, is consistent with the data-driven result of Eq. (8.3) but with a large enough uncertainty to also cover the “no new physics” scenario: 

a HVP, LO µ = 7116(184) × 10−11 , (8.6) 

based on Refs. [9–17]. 

The phenomenological estimate of HLbL scattering as reviewed in Sec. 4 is essentially based on a dispersive approach, in analogy to HVP. The various contributions to HLbL can be collected into three main pieces depending on how they have been estimated: (1) the numerically dominant contributions from the single-pseudoscalar poles and large parts of the two-pion intermediate states, both of which rely on data-driven approaches and are under good control; (2) the model-dependent estimates for the sum of scalar, tensor, and axial-vector contributions, as well as the impact of short-distance constraints; all of these still suffer from significant uncertainties, which in the total have been added linearly; (3) the c-quark contribution, which can be estimated using perturbative QCD, with a conservative uncertainty estimate in view of the low scale and potential nonperturbative effects. The final estimates for HLbL from Table 15 (mainly based on Refs. [18–30] and, in addition to e + e − → hadrons cross sections, the experimental input from Refs. [90–109]) and HLbL at NLO [31] from Eq. (4.91) read as follows: 

a HLbL µ = (69.3(4.1) + 20(19) + 3(1)) × 10−11 = 92(19) × 10−11 , (8.7) 

a HLbL, NLO µ = 2(1) × 10−11 , (8.8) 

where the first line gives the three pieces in the same order as discussed above and the total in the second line is obtained by adding the central values of the three contributions and combining the errors in quadrature. The final error is about 20% and is completely dominated by the model estimates of a numerically subdominant part of the total. 

The lattice determination of HLbL scattering is reviewed in Sec. 5. The lattice methodology for this quantity has advanced significantly in the last years [110–116] and has now reached a mature stage, resulting in a calculation [32] with reliable estimates of both statistical and systematic uncertainties (Eq. (5.49)): 

a HLbL µ = 78.7(30.6)stat(17.7)sys × 10−11 . (8.9) 

There have been extensive checks between different groups working on the lattice HLbL as well as internal checks of the calculations such as the regression against the leptonic loop or pion-pole contributions. These checks are explained in detail in Sec. 5. 

To obtain a recommendation for the full SM prediction we proceed as follows: for HLbL scattering, there is excellent agreement between phenomenology and lattice QCD, to the extent that it is justified to consider a weighted average. Taking into account that the lattice-QCD value does not include the c-quark loop, we first average the light-quark contribution and add the c quark as estimated phenomenologically in the end. This produces 

a HLbL µ (phenomenology + lattice QCD) = 90(17) × 10−11 , (8.10) 

and, using Eq. (8.8), 

a HLbL µ (phenomenology + lattice QCD) + a HLbL, NLO µ = 92(18) × 10−11 . (8.11) 

For HVP, the current uncertainties in lattice calculations are too large to perform a similar average and the future confrontation of phenomenology and lattice QCD crucially depends on the outcome of forthcoming lattice studies. For this reason, we adopt Eq. (8.3) as our final estimate, emphasizing that the uncertainty estimate already accounts for the tensions in the e + e − data base. Combined with the QED and EW contributions, we obtain 

a SM µ = a QED µ + a EW µ + a HVP, LO µ + a HVP, NLO µ + a HVP, NNLO µ + a HLbL µ + a HLbL, NLO µ = 116 591 810(43) × 10−11 . (8.12) 

This value is mainly based on Refs. [2–8, 18–24, 31–36], which should be cited in any work that uses or quotes Eq. (8.12). It differs from the Brookhaven measurement [1] 

a exp µ = 116 592 089(63) × 10−11 , (8.13) 

where the central value is adjusted to the latest value of λ = µµ/µp = 3.183345142(71) [751], by 

∆aµ := a exp µ − a SM µ = 279(76) × 10−11 , (8.14) 

corresponding to a 3.7σ discrepancy. In constructing Eqs. (8.5), (8.11), and (8.12), we have taken into account the correlations between the uncertainties in the leading and subleading HVP contributions as well as the partial correlation in the case of HLbL, with numbers rounded including subleading digits from the individual contributions. 

The prospects for near-term and long-term improvements of the uncertainties in the SM prediction are excellent. As discussed in Sec. 2, a new measurement of the crucial 2π channel by SND is currently under review, and more measurements of the 2π channel and others are forthcoming, leading to the realistic prospects of reducing the dispersive HVP error by a factor of 2. In addition, independent data-driven input could be provided by the MUonE project. The past five years have seen great progress in the development of methods to address the challenges associated with lattice determinations of a HVP, LO µ at the target precision, as discussed in detail in Sec. 3. This is also evident in the recent high-precision lattice result for a HVP, LO µ [392], which, however, still needs to be scrutinized in detail. With these methods now in place, and with sustained, dedicated effort, lattice results with permil-level precision will be forthcoming. The phenomenological determination of HLbL scattering has been consolidated at a level well below the Glasgow consensus, see Sec. 4, with the dominant contributions derived using data-driven methods in analogy to the dispersive HVP approach. With expected progress on the subleading contributions, a 10% calculation of HLbL scattering now appears feasible. Finally, we expect more independent lattice calculations of the HLbL to appear in the next years. Building on the newly developed methodologies, a 10% lattice calculation of the HLbL also appears feasible by the end of the Fermilab experiment.